Question

the sum of the digits of a two digit number is 12 .If 18 is added to it ,the digit are reversed .find the number​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

• Sum of the digits of a two digit number is 12.

So, Let assume that

• Digit at ones place be x

• Digit at tens place be 12 - x

So,

• Number formed = 10 × (12 - x) + x × 1 = 120 - 10x + x = 120 - 9x

And

• Reverse number = 10 × x + (12 - x) × 1 = 10x + 12 - x = 9x + 12

So, We have with us

$\begin{gathered}\begin{gathered}\bf\: \rm :\longmapsto\:\begin{cases} &\sf{digit \: at \: tens \: place \: be \: x} \\ \\ &\sf{digits \: at \: ones \: place \: be \: 12 - x} \end{cases}\end{gathered}\end{gathered}$

And

$\begin{gathered}\begin{gathered}\bf\: So-\begin{cases} &\sf{number \: formed = 120 - 9x} \\ \\ &\sf{reverse \: number = 12 + 9x} \end{cases}\end{gathered}\end{gathered}$

According to statement,

If 18 is added to the number, the digits are reversed

$\rm :\longmapsto\:120 - 9x + 18 = 9x + 12$

$\rm :\longmapsto\:138 - 9x= 9x + 12$

$\rm :\longmapsto\:- 9x - 9x= 12 - 138$

$\rm :\longmapsto\:- 18x= - 126$

$\bf\implies \:x = 7$

$\begin{gathered}\begin{gathered}\bf\: So-\begin{cases} &\sf{number \: formed = 120 - 9x = 120 - 63 = 57} \\ \\ &\sf{reverse \: number = 12 + 9x = 12 + 63 = 75} \end{cases}\end{gathered}\end{gathered}$

Hence, Two digit number is 57

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Basic Concept Used :-

Writing Systems of Linear Equation from Word Problem.

1. Understand the problem.

Understand all the words used in stating the problem.

Understand what you are asked to find.

2. Translate the problem to an equation.

Assign a variable (or variables) to represent the unknown.

Clearly state what the variable represents.

3. Carry out the plan and solve the problem.

Answer 2

GIVEN :

• the sum of the digits of a two digit number is 12 .
• If 18 is added to it ,the digit are reversed .

TO FIND :

• find the number

SOLUTION :

Let,

• The Ones digit be 'x'
• The number be '12x - x'

According to the question,

Solving,

$:\implies\sf \:10 x (12 - x) + × x 1$

$:\implies\sf \: 120 - 10x + x$

$:\implies\sf \:120 - 9x$

Reversed number,

$:\implies\sf \:10 × x + (12-x) x 1$

$:\implies\sf \:10x + 12 - x$

$:\implies\sf \: 9x + 12$

now let,

• digit at tens place be x
• digits at ones place be 12 - x

When 18 is added to the number, its digits are interchanged,

$:\implies\sf \:120 - 9x + 18 = 9x + 12$

$:\implies\sf \:138 - 9x = 9x + 12$

$:\implies\sf \:9x - 9x = 12-138$

$:\implies\sf \:-18x = -126$

$:\implies\sf \:x = 7$

number formed

• 120 - 9x = 120-63 = 57

• 12 + 9x = 12 + 63 = 75

The number is 57